In a complete space $X$ is every $x \in X$ the limit of a sequence $\{x_n\}$ such that $x \not\in \{x_n\}$?

Question

Let $X$ be a complete metric space. Then for any point $x \in X$, can it be shown that there exists a sequence $\{x_n\} \in X$ such that $x \not\in \{x_n\}$ and $x_n \rightarrow x$? More generally, I know that completeness implies every element of a complete space can be written as the limit of some sequence contained in the space, but how can this be shown?

Answer 1

No. The space which has only one point is a counterexample.